Step |
Hyp |
Ref |
Expression |
1 |
|
resco |
⊢ ( ( abs ∘ 𝐹 ) ↾ 𝐴 ) = ( abs ∘ ( 𝐹 ↾ 𝐴 ) ) |
2 |
|
o1f |
⊢ ( 𝐹 ∈ 𝑂(1) → 𝐹 : dom 𝐹 ⟶ ℂ ) |
3 |
|
lo1o1 |
⊢ ( 𝐹 : dom 𝐹 ⟶ ℂ → ( 𝐹 ∈ 𝑂(1) ↔ ( abs ∘ 𝐹 ) ∈ ≤𝑂(1) ) ) |
4 |
2 3
|
syl |
⊢ ( 𝐹 ∈ 𝑂(1) → ( 𝐹 ∈ 𝑂(1) ↔ ( abs ∘ 𝐹 ) ∈ ≤𝑂(1) ) ) |
5 |
4
|
ibi |
⊢ ( 𝐹 ∈ 𝑂(1) → ( abs ∘ 𝐹 ) ∈ ≤𝑂(1) ) |
6 |
|
lo1res |
⊢ ( ( abs ∘ 𝐹 ) ∈ ≤𝑂(1) → ( ( abs ∘ 𝐹 ) ↾ 𝐴 ) ∈ ≤𝑂(1) ) |
7 |
5 6
|
syl |
⊢ ( 𝐹 ∈ 𝑂(1) → ( ( abs ∘ 𝐹 ) ↾ 𝐴 ) ∈ ≤𝑂(1) ) |
8 |
1 7
|
eqeltrrid |
⊢ ( 𝐹 ∈ 𝑂(1) → ( abs ∘ ( 𝐹 ↾ 𝐴 ) ) ∈ ≤𝑂(1) ) |
9 |
|
fresin |
⊢ ( 𝐹 : dom 𝐹 ⟶ ℂ → ( 𝐹 ↾ 𝐴 ) : ( dom 𝐹 ∩ 𝐴 ) ⟶ ℂ ) |
10 |
|
lo1o1 |
⊢ ( ( 𝐹 ↾ 𝐴 ) : ( dom 𝐹 ∩ 𝐴 ) ⟶ ℂ → ( ( 𝐹 ↾ 𝐴 ) ∈ 𝑂(1) ↔ ( abs ∘ ( 𝐹 ↾ 𝐴 ) ) ∈ ≤𝑂(1) ) ) |
11 |
2 9 10
|
3syl |
⊢ ( 𝐹 ∈ 𝑂(1) → ( ( 𝐹 ↾ 𝐴 ) ∈ 𝑂(1) ↔ ( abs ∘ ( 𝐹 ↾ 𝐴 ) ) ∈ ≤𝑂(1) ) ) |
12 |
8 11
|
mpbird |
⊢ ( 𝐹 ∈ 𝑂(1) → ( 𝐹 ↾ 𝐴 ) ∈ 𝑂(1) ) |